Weak Solutions to the Navier–Stokes Inequality with Arbitrary Energy Profiles

Abstract

In a recent paper, Buckmaster & Vicol (arXiv:1709.10033) used the method of convex integration to construct weak solutions u to the 3D incompressible Navier-Stokes equations such that u(t) L 2 = e(t) for a given non-negative and smooth energy profile e : [0, T ] → R. However, it is not known whether it is possible to extend this method to construct nonunique solutions suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI), Leray-Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing.In this paper we are concerned with weak solutions to the Navier-Stokes inequality on R 3 , that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier-Stokes equations). Given T > 0 and a nonincreasing energy profile e : [0, T ] → [0, ∞) we construct weak solution to the Navier-Stokes inequality that are localised in space and whose energy profile u(t) L 2 (R 3 ) stays arbitrarily close to e(t) for all t ∈ [0, T ]. Our method applies only to nonincreasing energy profiles.The relevance of such solutions is that, despite not satisfying the Navier-Stokes equations, they satisfy the partial regularity theory of Caffarelli, Kohn & Nirenberg (Comm. Pure Appl. Math., 1982). In fact, Scheffer's constructions of weak solutions to the Navier-Stokes inequality with blow-ups (Comm. Math. Phys., 1985& 1987 show that the Caffarelli, Kohn & Nirenberg's theory is sharp for such solutions.Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer's. Namely, we obtain weak solutions to the Navier-Stokes inequality with both blow-up and a prescribed energy profile.

Partial Regularity for a Surface Growth Model

“…As with the Navier-Stokes equations, there is a gap here between the box-counting and Hausdorff dimensions; as with the NSE, it is an open question whether these dimension estimates can be equalised. In this context, it would be interesting to adapt the constructions due to Scheffer [21,22], see also Ożański [15,17]) of solutions of the weak form of the 'Navier-Stokes inequality' that have a space-time singular set of Hausdorff dimension γ for any γ ∈ (0, 1) to the SGM. This seems difficult, since the constructions make use of (i) the three-dimensional nature of the fluid flow and (ii) the pressure function plays a fundamental role in amplifying the magnitude of the velocity.…”

Convex integration and phenomenologies in turbulence

“…In view of the partial regularity results of Leray [128], Scheffer [168], the Navier-Stokes inequality results [171,172,159], and the recent work [15] it is natural to investigate the limits of partial regularity in the context of the Navier-Stokes equations. This leads us to following open problem.…”

From Jean Leray to the millennium problem: the Navier–Stokes equations

“…Since these solutions cannot be shown to satisfy the energy inequality on [T /3, 2T /3], the question arises whether there exist weak solutions in the sense of Leray-Hopf with given physically reasonable kinetic energy function. A partial answer is given by Ożański [48] who prescribes an arbitrary continuous, non-increasing energy function E kin (t) and constructs a suitable weak solution u satisfying the localized energy inequality such that 1 2 u(t) 2 2 − E kin (t) is bounded by any given positive constant. However, u satisfies a Navier-Stokes inequality, i.e., u is a solution in the Leray-Hopf class with the property that f := u t − νΔu + u • ∇u + ∇ p, f • u ≤ 0.…”